Animating active object in physics simulation
This has been prepared on the assumption that you're unfamiliar with Blender's animation and constraint handling...
(Blend file below)

When mixing animation, physics, parentage etc, a significant amount of time and trouble can be avoided by keeping things 'modular'.  i.e. get the first part working, bake it, then the second, bake that etc.
The philosophy behind this is because Blender's modifiers, physics, parentage etc, can all play havoc with each other when acting on the same model concurrently.
Strict and sometimes quite finnicky procedures can circumvent some of these interactions but only when adhered to, often to the letter, and assuming that the workarounds are even known!
The answer given here has been extended beyond what was asked for to illustrate how easy it can be to multi-switch between constraints and physics.   Picking up an object would usually require further action such as being carried off and placed somewhere else.  Mixing physics with similtaneous animation is also a relatively simple matter, but this project won't require that.
A hint - save your work with incremental filenames at every new stage!
Another -  Tick all the boxes in the baking panel.
See Below for the fine detail - re: baking frame ranges.
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The item to be picked up is the Monkey.  That is given Dynamic physics properties from frame 1 initially so it drops onto the conveyor.  When it settles the action can be baked from frame 1 up to the settling frame. (see details below)
Once baked that part of the action is set in concrete and won't change without direct intervention.  You have to manually switch off it's physics however and also delete the two physics strips left behind in the Monkey's Dopesheet entries.
With the physics gone, the Monkey can then use the conveyor as a parent (CTL-P) and that's what carries it along to the pick-up point.
Once reached, the parentage has to be switched to the pincer arm, so baking is used once again, but only for the necessary frames.  i.e. where the original parentage was introduced up to and including the parent change-over.
Two concurrent 'Child-of' constraints can do this, but in the long run, it's far far better to simply bake the action and preserve what's already working.
Once baked, the pincer arm can become the parent to both pick up and carry the Monkey.
It's lowered again and dropped gently.   Bake it again up to the drop frame.
Once baked it's free to be given physics again, where it drops, rolls, and falls off.  Bake that and your entire action is pretty much immune from future interference.
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Not one of these procedures encountered anything like the hassles that can crop up when keeping multiple constraints and physics active.
A year from now, the file can be reopened and guaranteed to perform exactly as before, working on the premise that a new version of Blender won't depart significantly from it's most basic animation handling-code.
Should any section need a change, that section only can be re-physic'd or whatever, then baked again.  (with all due care in the Dopesheet to accomodate the new action)
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Baking Frame ranges -
The initial drop onto the conveyor takes place between frames 1 to 47 and these are the frames to be baked.  Remember, if you bake an extra frame or 2 and it upsets the action, simply delete them in the Dopesheet.  (Save your work first)
The conveyor parentage, where it's carried along,  runs from frame 47 to 97.
The pincer arm parentage where it's picked up - frames 97 to 133.
Physics are applied again from frame 134 to 180 where it's rolls off and falls away.